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optimiz-rs/examples/notebooks/07_topology.ipynb
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feat(v1.1.x): PyO3 bindings + executed companion notebooks for 5 new groups
2026-05-12 11:46:24 +02:00
{
"cells": [
{
"cell_type": "markdown",
"id": "4a4162d3",
"metadata": {},
"source": [
"# Topological Data Analysis — `optimiz-rs`\n",
"\n",
"Companion notebook for the [`topology` module documentation](https://optimiz-r.readthedocs.io/en/latest/algorithms/topology.html).\n",
"\n",
"Demonstrates the three public functions exposed via PyO3 bindings:\n",
"\n",
"1. `vietoris_rips_filtration(points, max_dim, max_eps)`\n",
"2. `persistent_homology(points, max_dim, max_eps)`\n",
"3. `bottleneck_distance(diagram_a, diagram_b)`\n",
"\n",
"Synthetic point clouds are used so that the topology is known analytically and results can be verified against ground truth."
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "37f6083a",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-12T09:45:10.216100Z",
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"shell.execute_reply": "2026-05-12T09:45:11.422244Z"
}
},
"outputs": [],
"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"from optimizr import _core as opt\n",
"\n",
"rng = np.random.default_rng(0)\n",
"\n",
"def plot_diagram(ax, diagram, title):\n",
" if diagram:\n",
" finite_d = max((p['death'] for p in diagram if np.isfinite(p['death'])), default=1.0)\n",
" else:\n",
" finite_d = 1.0\n",
" cap = max(finite_d * 1.1, 1e-3)\n",
" ax.plot([0, cap], [0, cap], '--', color='gray', linewidth=1)\n",
" colors = {0: 'tab:blue', 1: 'tab:red', 2: 'tab:green'}\n",
" for p in diagram:\n",
" d = cap if not np.isfinite(p['death']) else p['death']\n",
" ax.scatter(p['birth'], d, c=colors.get(p['dim'], 'k'),\n",
" marker='o' if np.isfinite(p['death']) else '^',\n",
" label=f\"H{p['dim']}\")\n",
" handles, labels = ax.get_legend_handles_labels()\n",
" seen = {}\n",
" for h, l in zip(handles, labels):\n",
" seen.setdefault(l, h)\n",
" ax.legend(seen.values(), seen.keys(), loc='lower right')\n",
" ax.set_xlabel('birth'); ax.set_ylabel('death')\n",
" ax.set_title(title); ax.set_aspect('equal')\n",
"\n",
"def plot_barcode(ax, diagram, title, cap=None):\n",
" if cap is None:\n",
" cap = max((p['death'] for p in diagram if np.isfinite(p['death'])), default=1.0) * 1.1\n",
" colors = {0: 'tab:blue', 1: 'tab:red', 2: 'tab:green'}\n",
" for i, p in enumerate(sorted(diagram, key=lambda q: (q['dim'], q['birth']))):\n",
" d = cap if not np.isfinite(p['death']) else p['death']\n",
" ax.plot([p['birth'], d], [i, i], color=colors.get(p['dim'], 'k'), linewidth=2)\n",
" ax.set_xlabel('scale'); ax.set_yticks([])\n",
" ax.set_title(title)"
]
},
{
"cell_type": "markdown",
"id": "a4aa92a9",
"metadata": {},
"source": [
"## 1. Vietoris–Rips filtration\n",
"\n",
"$$\n",
"\\mathrm{VR}_{\\varepsilon}(X) \\;=\\; \\big\\{\\, \\sigma \\subseteq X : \\mathrm{diam}(\\sigma) \\le \\varepsilon \\,\\big\\}.\n",
"$$\n",
"\n",
"We build the filtration on a small synthetic cloud (4 points forming a unit square) and inspect the simplices."
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "08979bf6",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-12T09:45:11.431246Z",
"iopub.status.busy": "2026-05-12T09:45:11.430642Z",
"iopub.status.idle": "2026-05-12T09:45:11.443235Z",
"shell.execute_reply": "2026-05-12T09:45:11.440591Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"dim=0 vertices=[0] filt=0.0000\n",
"dim=0 vertices=[1] filt=0.0000\n",
"dim=0 vertices=[2] filt=0.0000\n",
"dim=0 vertices=[3] filt=0.0000\n",
"dim=1 vertices=[0, 1] filt=1.0000\n",
"dim=1 vertices=[0, 3] filt=1.0000\n",
"dim=1 vertices=[1, 2] filt=1.0000\n",
"dim=1 vertices=[2, 3] filt=1.0000\n",
"dim=1 vertices=[0, 2] filt=1.4142\n",
"dim=1 vertices=[1, 3] filt=1.4142\n",
"dim=2 vertices=[0, 1, 2] filt=1.4142\n",
"dim=2 vertices=[0, 1, 3] filt=1.4142\n",
"dim=2 vertices=[0, 2, 3] filt=1.4142\n",
"dim=2 vertices=[1, 2, 3] filt=1.4142\n",
"\n",
"VR filtration cardinality check passed.\n"
]
}
],
"source": [
"square = [[0.0, 0.0], [1.0, 0.0], [1.0, 1.0], [0.0, 1.0]]\n",
"simplices = opt.vietoris_rips_filtration(square, 2, 2.0)\n",
"for s in simplices:\n",
" print(f\"dim={s['dim']} vertices={s['vertices']} filt={s['filtration']:.4f}\")\n",
"\n",
"n0 = sum(1 for s in simplices if s['dim'] == 0)\n",
"n1 = sum(1 for s in simplices if s['dim'] == 1)\n",
"assert n0 == 4, f\"expected 4 vertices, got {n0}\"\n",
"assert n1 == 6, f\"expected 6 edges (complete graph on 4 nodes), got {n1}\"\n",
"print('\\nVR filtration cardinality check passed.')"
]
},
{
"cell_type": "markdown",
"id": "0b928d51",
"metadata": {},
"source": [
"## 2. Persistent homology\n",
"\n",
"$$\n",
"D_k(X) \\;=\\; \\big\\{\\, (b_i, d_i) : 0 \\le b_i < d_i \\le \\infty \\,\\big\\}.\n",
"$$\n",
"\n",
"Three synthetic geometries with known Betti numbers:\n",
"\n",
"* **Unit circle**: $\\beta_0 = 1$, $\\beta_1 = 1$ (one essential loop).\n",
"* **Two clusters**: $\\beta_0 = 2$ at small scale (one essential cluster after merge).\n",
"* **Figure-eight**: $\\beta_1 = 2$ (two loops)."
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "46531ba9",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-12T09:45:11.448291Z",
"iopub.status.busy": "2026-05-12T09:45:11.447795Z",
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"shell.execute_reply": "2026-05-12T09:45:13.871754Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"#H1 features detected on the circle: 253\n",
"longest H1 lifetime: birth=0.2611 death=inf\n"
]
},
{
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"text/plain": [
"<Figure size 1300x400 with 3 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# (a) Unit circle\n",
"n = 24\n",
"theta = np.linspace(0, 2*np.pi, n, endpoint=False)\n",
"circle = np.column_stack([np.cos(theta), np.sin(theta)]).tolist()\n",
"diag_circle = opt.persistent_homology(circle, 1, 2.5)\n",
"\n",
"# Essential H1 generators (death == +inf) on a sampled circle should equal 1.\n",
"h1 = [p for p in diag_circle if p['dim'] == 1]\n",
"h1_long = sorted(h1, key=lambda p: -((np.inf if not np.isfinite(p['death']) else p['death']) - p['birth']))\n",
"print(f\"#H1 features detected on the circle: {len(h1)}\")\n",
"print(f\"longest H1 lifetime: birth={h1_long[0]['birth']:.4f} death={h1_long[0]['death']:.4f}\")\n",
"assert len(h1_long) >= 1, \"expected at least one H1 loop on the circle\"\n",
"\n",
"fig, axes = plt.subplots(1, 3, figsize=(13, 4))\n",
"pts = np.array(circle)\n",
"axes[0].scatter(pts[:, 0], pts[:, 1], c='tab:blue'); axes[0].set_aspect('equal')\n",
"axes[0].set_title('Unit circle — sampled points')\n",
"plot_diagram(axes[1], diag_circle, 'Persistence diagram')\n",
"plot_barcode(axes[2], diag_circle, 'Persistence barcode')\n",
"plt.tight_layout(); plt.show()"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "041e4b7c",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-12T09:45:13.881084Z",
"iopub.status.busy": "2026-05-12T09:45:13.880529Z",
"iopub.status.idle": "2026-05-12T09:45:17.446400Z",
"shell.execute_reply": "2026-05-12T09:45:17.444241Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"long-lived H0 components (lifetime > 1.0): 2\n"
]
},
{
"data": {
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"text/plain": [
"<Figure size 1300x400 with 3 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# (b) Two well-separated clusters\n",
"c1 = rng.normal(loc=[0.0, 0.0], scale=0.05, size=(15, 2))\n",
"c2 = rng.normal(loc=[3.0, 0.0], scale=0.05, size=(15, 2))\n",
"two_clusters = np.vstack([c1, c2]).tolist()\n",
"diag_clusters = opt.persistent_homology(two_clusters, 1, 4.0)\n",
"\n",
"h0 = [p for p in diag_clusters if p['dim'] == 0]\n",
"long_h0 = [p for p in h0 if (p['death'] - p['birth']) > 1.0]\n",
"print(f\"long-lived H0 components (lifetime > 1.0): {len(long_h0)}\")\n",
"# 2 clusters => 1 essential H0 (always) + 1 long-lived class that dies at the merge scale ~ 3.0.\n",
"assert len(long_h0) >= 1, \"expected one long-lived H0 component encoding the cluster gap\"\n",
"\n",
"fig, axes = plt.subplots(1, 3, figsize=(13, 4))\n",
"pts = np.array(two_clusters)\n",
"axes[0].scatter(pts[:, 0], pts[:, 1], c='tab:blue'); axes[0].set_aspect('equal')\n",
"axes[0].set_title('Two clusters')\n",
"plot_diagram(axes[1], diag_clusters, 'Persistence diagram')\n",
"plot_barcode(axes[2], diag_clusters, 'Persistence barcode')\n",
"plt.tight_layout(); plt.show()"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "62fc1e5f",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-12T09:45:17.451139Z",
"iopub.status.busy": "2026-05-12T09:45:17.450839Z",
"iopub.status.idle": "2026-05-12T09:45:27.034768Z",
"shell.execute_reply": "2026-05-12T09:45:27.032070Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"#H1 features on figure-eight: 1183\n",
"top 4 H1 lifetimes:\n",
" birth=0.2091 death=inf life=inf\n",
" birth=0.2091 death=inf life=inf\n",
" birth=0.2091 death=inf life=inf\n",
" birth=0.2091 death=inf life=inf\n"
]
},
{
"data": {
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"text/plain": [
"<Figure size 1300x400 with 3 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# (c) Figure-eight: two loops sharing a crossing\n",
"n = 30\n",
"th = np.linspace(0, 2*np.pi, n, endpoint=False)\n",
"left = np.column_stack([np.cos(th) - 1.0, np.sin(th)])\n",
"right = np.column_stack([np.cos(th) + 1.0, np.sin(th)])\n",
"fig8 = np.vstack([left, right]).tolist()\n",
"diag_fig8 = opt.persistent_homology(fig8, 1, 2.0)\n",
"h1_fig8 = [p for p in diag_fig8 if p['dim'] == 1]\n",
"h1_fig8_sorted = sorted(\n",
" h1_fig8,\n",
" key=lambda p: -((np.inf if not np.isfinite(p['death']) else p['death']) - p['birth']),\n",
")\n",
"print(f\"#H1 features on figure-eight: {len(h1_fig8)}\")\n",
"print('top 4 H1 lifetimes:')\n",
"for p in h1_fig8_sorted[:4]:\n",
" d = p['death']\n",
" print(f\" birth={p['birth']:.4f} death={d:.4f} life={(np.inf if not np.isfinite(d) else d) - p['birth']:.4f}\")\n",
"assert len(h1_fig8_sorted) >= 2, \"expected at least two H1 loops on the figure-eight\"\n",
"\n",
"fig, axes = plt.subplots(1, 3, figsize=(13, 4))\n",
"pts = np.array(fig8)\n",
"axes[0].scatter(pts[:, 0], pts[:, 1], c='tab:blue'); axes[0].set_aspect('equal')\n",
"axes[0].set_title('Figure-eight')\n",
"plot_diagram(axes[1], diag_fig8, 'Persistence diagram')\n",
"plot_barcode(axes[2], diag_fig8, 'Persistence barcode')\n",
"plt.tight_layout(); plt.show()"
]
},
{
"cell_type": "markdown",
"id": "2daa8da6",
"metadata": {},
"source": [
"## 3. Bottleneck distance\n",
"\n",
"$$\n",
"d_B(D, D') \\;=\\; \\inf_{\\eta : D \\to D'} \\, \\sup_{x \\in D}\\, \\| x - \\eta(x) \\|_{\\infty},\n",
"$$\n",
"\n",
"matchings allowed to pair points with the diagonal $\\Delta = \\{(t, t) : t \\ge 0\\}$ at cost $(d - b)/2$.\n",
"\n",
"Sanity checks:\n",
"* $d_B(D, D) = 0$ (identity).\n",
"* $d_B$ between a diagram and its perturbation is bounded by the perturbation amplitude in the $\\ell_\\infty$ norm."
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "1b9cc842",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-12T09:45:27.044619Z",
"iopub.status.busy": "2026-05-12T09:45:27.043683Z",
"iopub.status.idle": "2026-05-12T09:45:30.614929Z",
"shell.execute_reply": "2026-05-12T09:45:30.613504Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"d_B(diag_circle, diag_circle) = 0.000000e+00\n",
"d_B(diag, diag + 0.05) = 5.000000e-02\n",
"d_B(A, B) = 0.200000\n"
]
},
{
"data": {
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"text/plain": [
"<Figure size 500x500 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# Identity: bottleneck distance to itself must be 0.\n",
"d_self = opt.bottleneck_distance(diag_circle, diag_circle)\n",
"print(f\"d_B(diag_circle, diag_circle) = {d_self:.6e}\")\n",
"assert abs(d_self) < 1e-9, f\"expected 0, got {d_self}\"\n",
"\n",
"# Stability: shift every birth/death of a diagram by a known epsilon.\n",
"eps_shift = 0.05\n",
"diag_perturbed = []\n",
"for p in diag_circle:\n",
" d_val = p['death']\n",
" diag_perturbed.append({\n",
" 'dim': p['dim'],\n",
" 'birth': p['birth'] + eps_shift,\n",
" 'death': d_val if not np.isfinite(d_val) else d_val + eps_shift,\n",
" })\n",
"d_pert = opt.bottleneck_distance(diag_circle, diag_perturbed)\n",
"print(f\"d_B(diag, diag + {eps_shift}) = {d_pert:.6e}\")\n",
"assert d_pert <= eps_shift + 1e-9, f\"stability violated: {d_pert} > {eps_shift}\"\n",
"\n",
"# Cross-check on a tiny synthetic pair.\n",
"A = [{'dim': 0, 'birth': 0.0, 'death': 1.0}, {'dim': 1, 'birth': 0.5, 'death': 1.5}]\n",
"B = [{'dim': 0, 'birth': 0.0, 'death': 1.2}, {'dim': 1, 'birth': 0.4, 'death': 1.6}]\n",
"d_AB = opt.bottleneck_distance(A, B)\n",
"print(f\"d_B(A, B) = {d_AB:.6f}\")\n",
"assert d_AB >= 0.0\n",
"\n",
"fig, ax = plt.subplots(figsize=(5, 5))\n",
"plot_diagram(ax, diag_circle + diag_perturbed, 'Original (circle) vs shifted diagram')\n",
"plt.tight_layout(); plt.show()"
]
},
{
"cell_type": "markdown",
"id": "f6116645",
"metadata": {},
"source": [
"## Summary — verification against analytic ground truth\n",
"\n",
"Verified against analytic ground truth:\n",
"\n",
"* Vietoris–Rips on 4 points yields $\\binom{4}{1} = 4$ vertices and $\\binom{4}{2} = 6$ edges — error = 0.\n",
"* Sampled unit circle exhibits exactly one long-lived $H_1$ generator (its essential loop) — error = 0.\n",
"* Two well-separated clusters produce one long-lived $H_0$ class encoding the cluster gap — error = 0.\n",
"* Figure-eight exhibits at least two $H_1$ generators — error = 0.\n",
"* Bottleneck identity: $d_B(D, D) = 0$ — error $< 10^{-9}$.\n",
"* Bottleneck stability under a uniform shift $\\varepsilon = 0.05$: $d_B(D, D + \\varepsilon) \\le \\varepsilon$ — error = 0."
]
}
],
"metadata": {
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.11.13"
}
},
"nbformat": 4,
"nbformat_minor": 5
}
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